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Globally solvable systems of complex vector fields. (English) Zbl 1242.35092
Summary: We consider a class of involutive systems of $$n$$ smooth vector fields on the $$n+1$$ dimensional torus. We obtain a complete characterization for the global solvability of this class in terms of Liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1-form in the minimal covering space.

##### MSC:
 35F05 Linear first-order PDEs 35N15 $$\overline\partial$$-Neumann problems and formal complexes in context of PDEs 35A30 Geometric theory, characteristics, transformations in context of PDEs
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##### References:
 [1] Bergamasco, A.P., Remarks about global analytic hypoellipticity, Trans. amer. math. soc., 351, 10, 4113-4126, (1999) · Zbl 0932.35046 [2] Bergamasco, A.P.; Cordaro, P.; Malagutti, P., Globally hypoelliptic systems of vector fields, J. funct. anal., 114, 267-285, (1993) · Zbl 0777.58041 [3] Bergamasco, A.P.; Cordaro, P.D.; Petronilho, G., Global solvability for certain classes of underdetermined systems of vector fields, Math. Z., 223, 261-274, (1996) · Zbl 0863.58062 [4] Bergamasco, A.P.; Kirilov, A., Global solvability for a class of overdetermined systems, J. funct. anal., 252, 603-629, (2007) · Zbl 1158.58011 [5] A.P. Bergamasco, A. Kirilov, W. Nunes, S.L. Zani, On the global solvability for overdetermined systems, Trans. Amer. Math. Soc., in press. · Zbl 1275.35004 [6] Bergamasco, A.P.; Nunes, W.; Zani, S.L., Global properties of a class of overdetermined systems, J. funct. anal., 200, 1, 31-64, (2003) · Zbl 1034.32024 [7] Bergamasco, A.P.; Petronilho, G., Global solvability of a class of involutive systems, J. math. anal. appl., 233, 314-327, (1999) · Zbl 0942.35011 [8] Berhanu, S.; Cordaro, P.; Hounie, J., An introduction to involutive structures, (2008), Cambridge University Press · Zbl 1151.35011 [9] Cardoso, F.; Hounie, J., Global solvability of an abstract complex, Proc. amer. math. soc., 65, 117-124, (1977) · Zbl 0335.58015 [10] Hounie, J., Globally hypoelliptic and globally solvable first-order evolution equations, Trans. amer. math. soc., 252, 233-248, (1979) · Zbl 0424.35030 [11] Treves, F., Study of a model in the theory of complexes of pseudodifferential operators, Ann. of math. (2), 104, 269-324, (1976) · Zbl 0354.35067 [12] Treves, F., Hypoanalytic structures (local theory), (1992), Princeton University Press Princeton, NJ · Zbl 0787.35003
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